Kiss those Math Headaches GOODBYE!

Archive for the ‘Philosophy’ Category

So here’s the situation: you’re at the breakfast table, enjoying a bowl of steaming-hot steel-cut oats and maple syrup, and you just poured yourself a mug of black coffee. But then you realize you want to pour some milk in the coffee (sorry, purists). But the milk is in the frig, six feet away. So of course you walk to the frig, grab the milk, bring it to the table, pour some in your coffee, return the milk to the frig and sit back down. Question: could you have done this more efficiently?

Yes, of course. You could have brought your cup of coffee with you as you walked to the frig, poured the milk right there at the frig, returned the milk, and then walked back to the table.

“Morning Joe”

When I realized this this morning, I thought … hmmm. Had I used a bit of forethought, I would save myself an entire round trip from the table to the frig. And while I have no problem making that extra trip (hey, just burned 1.3 calories, right?), the experience made me wonder if anyone has ever developed a mathematics of efficiency for running errands.

I could imagine someone taking initial steps for this. One would create symbols for the various aspects of errands. There would be a general symbol for an errand, and there would be a special ways of denoting: 1) an errand station (like the frig), 2) an errand that requires transporting an item (like carrying the mug), 3) an errand that requires doing an activity (pouring milk) with two items (mug and milk) at an errand station, 4) an errand that involves picking something up (picking up the mug), and so on. Then one could schematize the process and use it to code various kinds of errands. Eventually, perhaps, one could use such a system to analyze the most efficient way to, say, carry out 15 errands of which 3 involve transporting items, 7 involve picking things up, and 5 involve doing tasks at errand stations. Don’t get me wrong! I have not even begun to try this, but I’ve studied enough math that I can imagine it being done, and that’s one thing I love about math; it allows us to create general systems for analyzing real-world situations and thereby to do those activities more intelligently.

Of course, one reason I’m bringing this up is to encourage people to think more deeply about things that occur in their everyday lives. Activities that appear mundane can become mathematically intriguing when investigated. A wonderful example is the famous problem of the “Bridges of Konigsberg,” explored by the prolific mathematician Leonhard Euler nearly 300 years ago.

Euler in 1736 was living in the town of Konigsberg, now part of Russia. The Pregel River, which flows through Konigsberg, weaves around two islands that are part of the town, and a set of seven lovely bridges connect the islands to each other and to the town’s two river banks. For centuries Konigsberg’s residents wondered if there was a way to take a walk, starting at Point A, crossing each bridge exactly once, and return to Point A. But no one had found a way to do this.

One of the famous Seven Bridges of Konigsberg

Enter Euler. The great mathematician sat down and simplified the problem, turning the bridges into abstract line segments and transforming the bridge entrance and exits into points. Eventually Euler rigorously proved that there is no way to take the walk that people had wondered about. This would be just an interesting little tale, but it has a remarkable offshoot. After Euler published his proof, mathematicians took his way of simplifying the situation and, by exploring it, developed two new branches of math: topology and graph theory. The graph theory ideas that Euler first explored when thinking about the seven bridges sparked a branch of math that’s used today to determine the most efficient ways of connecting servers that form the backbone of the internet!

Of course, there’s also the classic example of Archimedes shouting “Eureka!” and running through the streets naked after seeing water rise in his bathtub. In that moment, Archimedes, who had been trying to help his king figure out if the crown that was just made for him had been created with pure gold, or with an alloy, saw that the water displacement would help him solve the problem. In the end, Archimedes determined that the crown was not pure gold, and the king rewarded the great thinker for his efforts.

As I write this, I find myself wondering if any of you readers can think of other situations in which everyday life experiences led mathematicians or scientists to major discoveries. It would be enlightening to hear more of these stories.

If you have a mathematical/logical bent of mind, you might find that interesting.

Friday the 13th is generally considered a bad luck day. So if that is the case, you might wonder if Monday the 13th would be the logical opposite to Friday the 13th, a good luck day. Afterall, Friday is the end of the workweek, and Monday is the beginning of the workweek.

So in that sense, can it be said that Monday and Friday are opposites? And what might that imply.

So here is the challenge. Compose a logical argument as to whether or not Monday the 13th should be considered a lucky day.

That is the challenge for Monday, the 13th of June 2011.

HINT: You may want to include information about the “truth value” (truthiness, as Steven Colbert likes to say) of statements and their converses.

REWARD: The first person who presents a compelling logical argument, one way or the other, wins a $10 gift certificate toward the purchase of any Singing Turtle Press products. All comments must be posted by 1 a.m. on Tuesday, the 14th of June, this year.

Amazon.com just listed the eBook TODAY, and so you can now get this book for just $9.95, and read it on ANY of these devices:

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The standard price of the paperback book is $19.95, but now you can get the eVersion of this book, and have it with you electronically, for HALF the price $9.95!

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The Algebra Survival Guide, which debuted in 2000, has sold more than a quarter of a million copies. It has also garnered a Parents Choice award, and it has been used in school districts all across the country as the cornerstone curriculum for Algebra Professional Development Workshops.

The book is read and used by struggling students, teachers, tutors, homeschoolers, and parents. It is an easy book to read, as it is written in a friendly Q&A conversational style. The companion workbook, soon to be available in an eVersion as well, provides thousands of additional practice problems.

Time for a post about geometry, which I tutor in addition to algebra and many other subjects.

I especially enjoy helping students learn how to do proofs, which I find is the hardest area of geometry for most kids.

Recently I came up with an analogy to help students understand the special usefulness of definitions in geometric proofs.

The analogy is: Definitions are like reversible coats.

What? … you say.

Coats. Reversible coats. As in two for the price of one.

Similarly with definitions: you get two IF-THEN statements for the price of one when you work with a definition.

Here’s what I mean.

First consider a “standard theorem” in geometry, viewed in the IF-THEN format.

Theorem: IF two angles are complements of the same angle, THEN they are congruent.

Notice that the converse of this statement doesn’t make much sense:

IF two angles are congruent, THEN they are complements of the same angle. (What other angle? We haven’t even mentioned another angle!)

But when it comes to definitions, you can:

a) First, turn the definition into and IF-THEN statement, and

b) Secondly, you can flip that IF-THEN statement around, and this new statement, called the “converse,” will always be true. You can bank on it!

Example of a definition: A right angle is an angle that measures 90 degrees.

And here’s one IF-THEN statement that flows out of this definition:

1) IF an angle is a right angle, THEN it measures 90 degrees.

But notice that the converse is also true:

2) IF an angle measures 90 degrees, THEN it is a right angle.

Let’s try this again, for the definition of perpendicular lines.

Definition: Two lines are perpendicular if they form four right angles.

First IF-THEN statement:

1) IF two lines are perpendicular, THEN they form four right angles.

Second IF-THEN statement, the converse.

2) IF two lines form four right angles, THEN the lines are perpendicular.

I am wondering if you are wondering why this is true. Why is it that, for definitions, both the statement and its converse are always true? The reason, I believe, has to do with the nature of a definition. With a definition, we are giving a name to some geometrical object, and stating what we consider to be the defining characteristic of that object.

To take a nonsensical example, suppose that you live in a world that has objects called “Snurfs,” which are measured in units called “Goobles.” Now imagine that some of the Snurfs are special because they have a measure of 100 Goobles. This fact makes these Snurfs so special that you wind up talking about them a lot. And because you talk about them a lot, it is helpful to give them a name. So you do give them a name; you decide to call them “Wombats.” What this means is that anytime a Snurf has a measure of 100 Goobles, you will call it a Wombat. And anytime you see the thing you call a Wombat, you can be sure that it will have a measure of 100 Goobles. For that is just what you have decided the word Wombat will mean. Based on this, you put forth the formal definition:

A Wombat is a Snurf with a measure of 100 Goobles.

Given this definition, notice that you can create two IF-THEN statements:

1) IF a Snurf is a Wombat, THEN its measure is 100 Goobles.

And you can also state the converse, and it will be true:

2) IF a Snurf has a measure of 100 Goobles, THEN it is a Wombat.

To me, this is how definitions work. They involve people noticing something they are talking about, and they decide to give it a name so they can talk about it more easily. When they define what the word means, they attach the word to the primary characteristic of this thing, and through this act, the word is born, and along with it, its definition.

Image via Wikipedia

Anyhow, in terms of doing geometry, the important thing to keep in mind is that all definitions can be used reversibly. So, going back to the example of the right angle, here’s what this means.

If, in the course of a proof, you establish that a particular angle is a right angle, you can conclude that the measure of this angle is 90 degrees. Reason: Definition of a right angle.

And similarly, if in a proof you establish that a particular angle has a measure of 90 degrees, then you can conclude that this angle is a right angle. Reason: Definition of a right angle.

This reversibility factor is why, when you read through geometric proofs, you will notice that “Definition of … ” is used quite often as a reason for steps. Because they are logically reversible, definitions are TWICE as useful as standard theorems.

Here’s an idea I came up with today for helping students understand more deeply the mistakes they make in algebra.

One thing that makes algebra difficult is that students have, basically, no sense as to whether something is true — or not — when they look at algebra. They have virtually no intuition about this. However, they do have intuition as to whether or not things are correct in arithmetic.

But we can use this idea to help students understand algebra. For example, we can use this approach to help students understand what is “wrong” when they make mistakes in algebra.

For example, let’s say that a student makes the following mistake:

3x – 4 – 2x = 12

+ 2x + 2x

5x – 4 = 12

What is the student is doing wrong? The student is adding 2x to the same side of the equation two times, instead of adding it to both sides of the equation.

How can we help the student see that this is wrong?

Change it to an arithmetic situation. Ask them is the following makes sense:

9 + 3 = 12

– 3 – 3

6 = 12

They will see that this is wrong because they know that the addition is wrong. What is more, they will get the general idea that it makes no sense to subtract 3 twice from the same side of the equation.

This mistake — in the algebra — will make little sense unless you do something like this, something they can grasp.

It’s sort of the Pink Elephant that has landed in the living room, and I can’t just pretend it’s not here any more

The election of Barack Obama could portend significant changes in the system of public education in this country.

If you’re open to sharing, I’d really like to hear your thoughts.

What do you hope for in an Obama Administration, with regard to education? Research and new Grants? Changes to No Child Left Behind? Greater funding for urban and rural districts? Higher teacher pay? An end to vouchers? More accountability? Merit pay?

What are your concerns?

Do you have any ideas that you think Obama would do well to heed?

What might Obama ignore in the field of education that he would do well to pay close attention to?

I’m opening this up pretty wide. But with one restriction. All comments must be on the topic of education. Any that are not on education I will have to discard.

Your coments may be short, Just, what’s on the top of your your mind? I’d really like to know. And we can all benefit by hearing from one anoher.

Like this:

If your math class is snoozing, try exploring the paradoxes that math touches on. One way is to just listen carefully to student’s questions, and see if there are any “big ideas” hidden in the question. Here’s an example of such an experience.

Recently I was tutoring a girl on the concept of rounding off decimals. This might sound dull, but this girl asked a question that, if you think about it, touches on the concept of infinity.

The student was looking at the number line and noticing how her textbook enlarged one small segment on it to display the problem, which was to round 2.72 to the nearest tenth.

The textbook’s number line took a “magnifier-approach,” blowing up only the section from 2.7 to 2.8, but showing all of the hundredth’s places in between.

The girl took a hard look at that and said, “So couldn’t you also take the space between something like 2.73 and 2.74, and blow that up?”

I asked her to explain. She said that if this smaller space were also ‘blown up’ or expanded, the new number line would display even smaller numbers, like 2.731, 2.732, 2.733, with 2.735 in the middle.

I told her that you could do this.

Then she asked, “Couldn’t you go even further?” Meaning, it turns out, can you then take an ever smaller part of the number line, such as the space from 2.731 and 2.732, and blow up that space?

I said you could.

She said, “Can you just keep doing it forever?”

I said you could.

She paused, then said: “I just don’t get that idea of ‘forever.'”

That was the moment …

I said it’s a hard idea to understand forever. But it’s an interesting thing to think about. I didn’t mention it to her, but later I realized that this 4th grader was essentially intrigued by the same question that captivated the mathematician/philosopher Blaise Pascal back about 300 years ago. Namely that humans are surrounded by two different infinities: the infinity of hugeness and the infinity of smallness. For more on Pascal’s words on this amazing matter, see http://www.leibniz-translations.com/pascal.htm

In any case, following this girl’s thought led her and me into a whole discussion about forever and infinity and the edge of the universe, and the conversation would have literally gone on “forever,” but eventually it was time to stop.

In our daily attempts to teach math we sometimes neglect to mention that math touches on the infinite, as an asymptote approaches a curve, you might say. Taking time, once in a while, to explore the infinite can make your class or tutoring session come very much alive.

And if you’d like to see a book that encourages “edgy” math questions by both students and teachers, check this out: http://www.amazon.com/Good-Questions-Math-Teaching-Them/dp/0941355519 It’s a book devoted to generating and listening to startling math questions. And it shows how these questions take a jackhammer to old musty classrooms, letting the light of curisoity and exploration get their day in the sun.

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